On 03/14/2006 01:54 AM, Gabriel Dos Reis wrote:
"Bill Page" <[EMAIL PROTECTED]> writes:
| On March 13, 2006 6:34 AM Ralf Hemmecke asked:
| > ...
| > But here the question to our category theory experts:
| > Since Monoid is something like (*,1) would it make sense
| > to speak of a category (in the mathematical sense) of
| >
| > monoids that have * and 1 as their operations
| > (1)
| >
| > ? Morphisms would respect 1 not just the identity element
| > with respect to *. And for each morphism f we would have
| > f(a*b) = f(a)*f(b). Of course as operations the two * above
| > are different but in that category they have to have the same
| > name. (No idea whether this makes sense, but it seems that
| > this is the way as "Category" it is implemented in Axiom/Aldor.)
| >
| > Then, of course, (N, +, 0) is not an object in the category
| > given by (1).
| >
|
| I keep trying to answer these questions but I am not sure I
| would like to classify myself as an "expert" in category theory.
| :) But here goes ...
|
| In category theory **Mon** (** means written in bold face font)
| consists of all monoids (as objects) and all monoid homomorphisms
| as morphisms. This does not say anything directly about what
| operation are present "inside" the objects of the category.
Agreed.
Well, Bill, are saying that I cannot have (1) as a category? That a
monoid itself is a category is true, but distracts from my question.
A category is a collection of objects and morphisms. Let's define MYMON.
Let an object be as follows:
A set M together with to operations *: (M, M) -> M and 1: M such that
the monoid axioms are satisfied.
(I am lazy and don't specify them here explicitly.)
The morphisms in MYMON are the usual monoid-homomorphisms.
And yes, if something wants to be an object in MYMON, it has to have *
as its binary operation and not +.
Who wouldn't one consider MYMON as a category?
And yes, I consider (N, +, 0) not an object in MYMON. What I try to say
is that Aldor has currently this view for the "Monoid" category.
Ralf
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